In papers [1]}, [2]},[3]} and [4]}, Joyce and collaborators have developed a rigorous cohomological Donaldson-Thomas theory, using the language of \([-1]\)  shifted symplectic structures in derived geometry introduced in [5]}. Informally, derived geometry replaces the notion of tangent space \(T_X\)  by a tangent complex \(\mathbb {T}_X\) , where \((\mathbb {T}_X)_0\)  gives the tangent directions, \((\mathbb {T}_X)_1\)  gives the obstructions, \((\mathbb {T}_X)_2\) , \((\mathbb {T}_X)_3\) , ... give the higher obstructions, and \((\mathbb {T}_X)_{-1}\)  gives the infinitesimal automorphisms. A \([-1]\) -shifted symplectic structure gives a pairing between \(\mathbb {T}_X\)  and \(\mathbb {T}_X[-1]\) : in particular, it pairs tangent directions with obstructions. In [5]}, it was shown that moduli stack of complexes on a Calabi-Yau 3-fold, and intersections of two Lagrangians in a symplectic space, carry naturally a \([-1]\) -shifted symplectic structure.
